Theorem prnz 4735

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4720	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4296	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  ∅c0 4285  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4582  df-pr 4584

This theorem is referenced by:  opnz  5440  propssopi  5476  fiint  9267  wilthlem2  27110  upgrbi  29240  wlkvtxiedg  29771  shincli  31511  chincli  31609  constrextdg2lem  34006  spr0nelg  48046  sprvalpwn0  48053